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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math3.analysis.integration.gauss;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import org.apache.commons.math3.util.Pair;<a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math3.exception.NotStrictlyPositiveException;<a name="line.20"></a>
<FONT color="green">021</FONT>    import org.apache.commons.math3.exception.util.LocalizedFormats;<a name="line.21"></a>
<FONT color="green">022</FONT>    <a name="line.22"></a>
<FONT color="green">023</FONT>    /**<a name="line.23"></a>
<FONT color="green">024</FONT>     * Factory that creates Gauss-type quadrature rule using Legendre polynomials.<a name="line.24"></a>
<FONT color="green">025</FONT>     * In this implementation, the lower and upper bounds of the natural interval<a name="line.25"></a>
<FONT color="green">026</FONT>     * of integration are -1 and 1, respectively.<a name="line.26"></a>
<FONT color="green">027</FONT>     * The Legendre polynomials are evaluated using the recurrence relation<a name="line.27"></a>
<FONT color="green">028</FONT>     * presented in &lt;a href="http://en.wikipedia.org/wiki/Abramowitz_and_Stegun"<a name="line.28"></a>
<FONT color="green">029</FONT>     * Abramowitz and Stegun, 1964&lt;/a&gt;.<a name="line.29"></a>
<FONT color="green">030</FONT>     *<a name="line.30"></a>
<FONT color="green">031</FONT>     * @since 3.1<a name="line.31"></a>
<FONT color="green">032</FONT>     * @version $Id: LegendreRuleFactory.java 1382197 2012-09-07 22:35:01Z erans $<a name="line.32"></a>
<FONT color="green">033</FONT>     */<a name="line.33"></a>
<FONT color="green">034</FONT>    public class LegendreRuleFactory extends BaseRuleFactory&lt;Double&gt; {<a name="line.34"></a>
<FONT color="green">035</FONT>        /**<a name="line.35"></a>
<FONT color="green">036</FONT>         * {@inheritDoc}<a name="line.36"></a>
<FONT color="green">037</FONT>         */<a name="line.37"></a>
<FONT color="green">038</FONT>        @Override<a name="line.38"></a>
<FONT color="green">039</FONT>        protected Pair&lt;Double[], Double[]&gt; computeRule(int numberOfPoints)<a name="line.39"></a>
<FONT color="green">040</FONT>            throws NotStrictlyPositiveException {<a name="line.40"></a>
<FONT color="green">041</FONT>            if (numberOfPoints &lt;= 0) {<a name="line.41"></a>
<FONT color="green">042</FONT>                throw new NotStrictlyPositiveException(LocalizedFormats.NUMBER_OF_POINTS,<a name="line.42"></a>
<FONT color="green">043</FONT>                                                       numberOfPoints);<a name="line.43"></a>
<FONT color="green">044</FONT>            }<a name="line.44"></a>
<FONT color="green">045</FONT>    <a name="line.45"></a>
<FONT color="green">046</FONT>            if (numberOfPoints == 1) {<a name="line.46"></a>
<FONT color="green">047</FONT>                // Break recursion.<a name="line.47"></a>
<FONT color="green">048</FONT>                return new Pair&lt;Double[], Double[]&gt;(new Double[] { 0d },<a name="line.48"></a>
<FONT color="green">049</FONT>                                                    new Double[] { 2d });<a name="line.49"></a>
<FONT color="green">050</FONT>            }<a name="line.50"></a>
<FONT color="green">051</FONT>    <a name="line.51"></a>
<FONT color="green">052</FONT>            // Get previous rule.<a name="line.52"></a>
<FONT color="green">053</FONT>            // If it has not been computed yet it will trigger a recursive call<a name="line.53"></a>
<FONT color="green">054</FONT>            // to this method.<a name="line.54"></a>
<FONT color="green">055</FONT>            final Double[] previousPoints = getRuleInternal(numberOfPoints - 1).getFirst();<a name="line.55"></a>
<FONT color="green">056</FONT>    <a name="line.56"></a>
<FONT color="green">057</FONT>            // Compute next rule.<a name="line.57"></a>
<FONT color="green">058</FONT>            final Double[] points = new Double[numberOfPoints];<a name="line.58"></a>
<FONT color="green">059</FONT>            final Double[] weights = new Double[numberOfPoints];<a name="line.59"></a>
<FONT color="green">060</FONT>    <a name="line.60"></a>
<FONT color="green">061</FONT>            // Find i-th root of P[n+1] by bracketing.<a name="line.61"></a>
<FONT color="green">062</FONT>            final int iMax = numberOfPoints / 2;<a name="line.62"></a>
<FONT color="green">063</FONT>            for (int i = 0; i &lt; iMax; i++) {<a name="line.63"></a>
<FONT color="green">064</FONT>                // Lower-bound of the interval.<a name="line.64"></a>
<FONT color="green">065</FONT>                double a = (i == 0) ? -1 : previousPoints[i - 1].doubleValue();<a name="line.65"></a>
<FONT color="green">066</FONT>                // Upper-bound of the interval.<a name="line.66"></a>
<FONT color="green">067</FONT>                double b = (iMax == 1) ? 1 : previousPoints[i].doubleValue();<a name="line.67"></a>
<FONT color="green">068</FONT>                // P[j-1](a)<a name="line.68"></a>
<FONT color="green">069</FONT>                double pma = 1;<a name="line.69"></a>
<FONT color="green">070</FONT>                // P[j](a)<a name="line.70"></a>
<FONT color="green">071</FONT>                double pa = a;<a name="line.71"></a>
<FONT color="green">072</FONT>                // P[j-1](b)<a name="line.72"></a>
<FONT color="green">073</FONT>                double pmb = 1;<a name="line.73"></a>
<FONT color="green">074</FONT>                // P[j](b)<a name="line.74"></a>
<FONT color="green">075</FONT>                double pb = b;<a name="line.75"></a>
<FONT color="green">076</FONT>                for (int j = 1; j &lt; numberOfPoints; j++) {<a name="line.76"></a>
<FONT color="green">077</FONT>                    final int two_j_p_1 = 2 * j + 1;<a name="line.77"></a>
<FONT color="green">078</FONT>                    final int j_p_1 = j + 1;<a name="line.78"></a>
<FONT color="green">079</FONT>                    // P[j+1](a)<a name="line.79"></a>
<FONT color="green">080</FONT>                    final double ppa = (two_j_p_1 * a * pa - j * pma) / j_p_1;<a name="line.80"></a>
<FONT color="green">081</FONT>                    // P[j+1](b)<a name="line.81"></a>
<FONT color="green">082</FONT>                    final double ppb = (two_j_p_1 * b * pb - j * pmb) / j_p_1;<a name="line.82"></a>
<FONT color="green">083</FONT>                    pma = pa;<a name="line.83"></a>
<FONT color="green">084</FONT>                    pa = ppa;<a name="line.84"></a>
<FONT color="green">085</FONT>                    pmb = pb;<a name="line.85"></a>
<FONT color="green">086</FONT>                    pb = ppb;<a name="line.86"></a>
<FONT color="green">087</FONT>                }<a name="line.87"></a>
<FONT color="green">088</FONT>                // Now pa = P[n+1](a), and pma = P[n](a) (same holds for b).<a name="line.88"></a>
<FONT color="green">089</FONT>                // Middle of the interval.<a name="line.89"></a>
<FONT color="green">090</FONT>                double c = 0.5 * (a + b);<a name="line.90"></a>
<FONT color="green">091</FONT>                // P[j-1](c)<a name="line.91"></a>
<FONT color="green">092</FONT>                double pmc = 1;<a name="line.92"></a>
<FONT color="green">093</FONT>                // P[j](c)<a name="line.93"></a>
<FONT color="green">094</FONT>                double pc = c;<a name="line.94"></a>
<FONT color="green">095</FONT>                boolean done = false;<a name="line.95"></a>
<FONT color="green">096</FONT>                while (!done) {<a name="line.96"></a>
<FONT color="green">097</FONT>                    done = b - a &lt;= Math.ulp(c);<a name="line.97"></a>
<FONT color="green">098</FONT>                    pmc = 1;<a name="line.98"></a>
<FONT color="green">099</FONT>                    pc = c;<a name="line.99"></a>
<FONT color="green">100</FONT>                    for (int j = 1; j &lt; numberOfPoints; j++) {<a name="line.100"></a>
<FONT color="green">101</FONT>                        // P[j+1](c)<a name="line.101"></a>
<FONT color="green">102</FONT>                        final double ppc = ((2 * j + 1) * c * pc - j * pmc) / (j + 1);<a name="line.102"></a>
<FONT color="green">103</FONT>                        pmc = pc;<a name="line.103"></a>
<FONT color="green">104</FONT>                        pc = ppc;<a name="line.104"></a>
<FONT color="green">105</FONT>                    }<a name="line.105"></a>
<FONT color="green">106</FONT>                    // Now pc = P[n+1](c) and pmc = P[n](c).<a name="line.106"></a>
<FONT color="green">107</FONT>                    if (!done) {<a name="line.107"></a>
<FONT color="green">108</FONT>                        if (pa * pc &lt;= 0) {<a name="line.108"></a>
<FONT color="green">109</FONT>                            b = c;<a name="line.109"></a>
<FONT color="green">110</FONT>                            pmb = pmc;<a name="line.110"></a>
<FONT color="green">111</FONT>                            pb = pc;<a name="line.111"></a>
<FONT color="green">112</FONT>                        } else {<a name="line.112"></a>
<FONT color="green">113</FONT>                            a = c;<a name="line.113"></a>
<FONT color="green">114</FONT>                            pma = pmc;<a name="line.114"></a>
<FONT color="green">115</FONT>                            pa = pc;<a name="line.115"></a>
<FONT color="green">116</FONT>                        }<a name="line.116"></a>
<FONT color="green">117</FONT>                        c = 0.5 * (a + b);<a name="line.117"></a>
<FONT color="green">118</FONT>                    }<a name="line.118"></a>
<FONT color="green">119</FONT>                }<a name="line.119"></a>
<FONT color="green">120</FONT>                final double d = numberOfPoints * (pmc - c * pc);<a name="line.120"></a>
<FONT color="green">121</FONT>                final double w = 2 * (1 - c * c) / (d * d);<a name="line.121"></a>
<FONT color="green">122</FONT>    <a name="line.122"></a>
<FONT color="green">123</FONT>                points[i] = c;<a name="line.123"></a>
<FONT color="green">124</FONT>                weights[i] = w;<a name="line.124"></a>
<FONT color="green">125</FONT>    <a name="line.125"></a>
<FONT color="green">126</FONT>                final int idx = numberOfPoints - i - 1;<a name="line.126"></a>
<FONT color="green">127</FONT>                points[idx] = -c;<a name="line.127"></a>
<FONT color="green">128</FONT>                weights[idx] = w;<a name="line.128"></a>
<FONT color="green">129</FONT>            }<a name="line.129"></a>
<FONT color="green">130</FONT>            // If "numberOfPoints" is odd, 0 is a root.<a name="line.130"></a>
<FONT color="green">131</FONT>            // Note: as written, the test for oddness will work for negative<a name="line.131"></a>
<FONT color="green">132</FONT>            // integers too (although it is not necessary here), preventing<a name="line.132"></a>
<FONT color="green">133</FONT>            // a FindBugs warning.<a name="line.133"></a>
<FONT color="green">134</FONT>            if (numberOfPoints % 2 != 0) {<a name="line.134"></a>
<FONT color="green">135</FONT>                double pmc = 1;<a name="line.135"></a>
<FONT color="green">136</FONT>                for (int j = 1; j &lt; numberOfPoints; j += 2) {<a name="line.136"></a>
<FONT color="green">137</FONT>                    pmc = -j * pmc / (j + 1);<a name="line.137"></a>
<FONT color="green">138</FONT>                }<a name="line.138"></a>
<FONT color="green">139</FONT>                final double d = numberOfPoints * pmc;<a name="line.139"></a>
<FONT color="green">140</FONT>                final double w = 2 / (d * d);<a name="line.140"></a>
<FONT color="green">141</FONT>    <a name="line.141"></a>
<FONT color="green">142</FONT>                points[iMax] = 0d;<a name="line.142"></a>
<FONT color="green">143</FONT>                weights[iMax] = w;<a name="line.143"></a>
<FONT color="green">144</FONT>            }<a name="line.144"></a>
<FONT color="green">145</FONT>    <a name="line.145"></a>
<FONT color="green">146</FONT>            return new Pair&lt;Double[], Double[]&gt;(points, weights);<a name="line.146"></a>
<FONT color="green">147</FONT>        }<a name="line.147"></a>
<FONT color="green">148</FONT>    }<a name="line.148"></a>




























































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